Finite Group Databasebeta

← #27296888

Subgroups — #27296888

Order 13568.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280380C211yesmaximal, characteristic
2#20273740C212yesmaximal
3#20272738C212yesmaximal
4#20272645C211yesmaximal, characteristic
5C53×#2030C211yesmaximal, characteristic
6#16810956—22no—
7#16810559C2212yes—
8#16810535—21no—
9#16810527—22no—
10#16810510—22no—
11C4×C22⋊Dic53—21no—
12#16810360—22no—
13#16810347C2212yes—
14C53×#543C2211yescharacteristic
15C53×#381—21no—
16C53×#381C2211yescharacteristic
17C53×#342—21no—
18C53×#342C2211yescharacteristic
19C22×Dic106—21no—
20C2×C4⋊Dic53D412yes—
21C2×#11719919—22no—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—21no—
24C2×C4×Dic53—21no—
25#13133150—42no—
26#13133141—42no—
27C22⋊#88322—22no—
28#13133124—21no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C23×C4×C53D411yescharacteristic
33C22⋊Q8×C53—22no—
34C22⋊Q8×C53—42no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—42no—
41C24×C53C2×D411yescharacteristic
42C2×Q8×C53—22no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—22no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53Q1612yes—
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C2×Dic106—44no—
55C2×Dic106—81no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—41no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C22⋊C4×C53—41no—
64C22⋊Dic53—44no—
65C4⋊Dic53—42no—
66#11719919—44no—
67C4×Dic53—81no—
68C4×Dic53—81no—
69C2×#88322—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—42no—
73C23×C53#12011yescharacteristic
74C23×C53C2×Q1611yescharacteristic
75C23×C53#13711yescharacteristic
76Q8×C53—82no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—81no—
87C2×C4×C53—82no—
88C2×C4×C53—81no—
89C2×C4×C53—82no—
90C2×Dic53—24no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96Dic106—84no—
97#88322—82no—
98#12216—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—81no—
106C22×C53#45111yescharacteristic
107C22×C53#45511yescharacteristic
108C22×C53B(4,𝔽2)11yescharacteristic
109C4×C53—42no—
110C4×C53—42no—
111C4×C53—81no—
112C4×C53—82no—
113C4×C53—161no—
114Dic53—44no—
115Dic53—82no—
116#2607—531no—
117#2030D5311yescharacteristic
118#1614—532no—
119#1352—532no—
120#1317—531no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—41no—
124C2×C53—41no—
125C2×C53#160211yescharacteristic
126C2×C53#219812yes—
127#543C2×D5311yescharacteristic
128#484—1062no—
129#399—532no—
130#393—1061no—
131#391—1062no—
132#387—1062no—
133#381—21no—
134#381C2×D5311yescharacteristic
135C4×C22⋊C4—1061no—
136#342—21no—
137#342C2×D5311yescharacteristic
138#328—1062no—
139#323—532no—
140C53#1221611yescharacteristic, complemented, Sylow
141C22×Q8—1061no—
142C23×C4C22⋊D5311yescharacteristic
143C22⋊Q8—22no—
144C22⋊Q8—42no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—1062no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4C22⋊D5311yescharacteristic
151C2×C22⋊C4C22×D5311yescharacteristic
152C2×C42—1061no—
153C2×C42—1061no—
154#107—2122no—
155#103—2122no—
156C22⋊C8—1062no—
157#95—22no—
158#95—42no—
159#95—532no—
160#95—1062no—
161#95—1061no—
162#95—1062no—
163C24C2×C22⋊D5311yescharacteristic
164C2×Q8—22no—
165C2×Q8—4241no—
166C2×Q8—2124no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—1062no—
175C22×C4—1062no—
176C22×C4#1171992512yes—
177C22×C4D4×D5312yes—
178C2×C8—2122no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—2122no—
182C4⋊C4—2124no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—2124no—
190C42—4241no—
191C42—4241no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23#1313318511yescharacteristic
196C23C2×#1171992511yescharacteristic
197C23#1313324711yescharacteristic
198Q8—82no—
199Q8—4244no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—1064no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2122no—
219C8—4242no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22#1681077811yescharacteristic
228C22#1681080011yescharacteristic
229C22#1681080911yescharacteristic
230C4—42no—
231C4—42no—
232C4—81no—
233C4—82no—
234C4—161no—
235C4—4242no—
236C4—2124no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2#2027367111yescharacteristic
242C2#2027670912yes—
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2030C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2030D53
hypocenterC53#12216
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
140. C53#12216semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202803802
#202737402
#202727382
#202726452
C53×#20302
#1221653